Category Archives: Examples

What is Scientific Notation?

Scientific notation is a style of writing numbers that are too large or too small to be expressed well by the common decimal notation. In scientific notation the decimal place is moved until only one number appears on the left side of the decimal. The number of places the decimal had to be moved is recorded using an exponent above a base of ten. (The terms standard form or exponential notation also refer to scientific notation.)

A typical decimal number expressed in scientific notation

Why Use Scientific Notation?

When dealing with numbers that are either very small or very large it can be difficult to express these in standard decimal notation. For example the number one billion 1,000,000,000 has many zeros. In science and engineering particularly even larger or smaller numbers occur often and writing so many zeros would be difficult and could lead to errors.

Examples of some typical decimal numbers in scientific notation

Okay so in the numbers above moving the decimal isn’t a huge headache saving strategy, but as numbers grow or shrink much further this new style of notation becomes essential.

How to Convert a Decimal to Scientific Notation:

Move the decimal left or right as many places as required until there is only one number to the left of the decimal.

The number of places you had to move the decimal is your exponent.

The direction you had to move the decimal, left or right, determines if your exponent will be positive or negative.

Moving the decimal left means a positive exponent.

Moving the decimal right means a negative exponent.

We always want to move the decimal left or right one whole spot at a time, because our numbering system is base ten this means the base of the exponent must be ten. Check the example below for the final rewrite.

Examples of Converting a Decimal Number to Scientific Notation

a. In the first example above the decimal is moved three places to the left. The decimal number 5780 can be rewritten as 5.78 x 10^3. Three is the exponent because this is how many places the decimal had to move. The exponent is positive because the decimal moved to the left.

b. In the second example above the decimal moves two places to the right. The decimal number 0.034 can be rewritten as 3.4 x 10^-2. The decimal moves two spaces so this is the exponent and it moves right so the exponent is negative.

How to Convert Scientific Notation to Decimal:

Move the decimal the number of places indicated by the exponent.

Again, the direction you’ll move the decimal depends on the exponent being positive or negative. When converting from scientific notation to decimal move in the opposite direction. Positive exponents move the decimal right and negative exponents move the decimal left.

Examples of Converting Scientific Notation to Decimal

a. In the first example our number in scientific notation is converted back into a decimal. The exponent is three so the decimal must move three places. In this case we are going from scientific notation to decimal so move the decimal place right for a positive exponent. Note: You can always approach the problem literally. 10^3 is the same as 1000. 5.78 x 1000 is 5780, the correct answer.

b. In the second example we can see the decimal must be moved two places. The numbers is changing from scientific notation to decimal so a negative decimal place indicates we must move the decimal to the left.

Scientific Notation on Calculators and Computers

Calculators and computers need a special way to represent scientific notation. You may have already seen this special notation used on really really large numbers in the online calculator. On calculators especially placing a power floating in the air above the line is too challenging so instead an exponent is represent with the letter “E”. To save even more space the multiplication symbol and base of ten are dropped completely and everything is simply replaced by “E”.

For example 5.78E3 or 5.78e3 are the same as 5.78 x 10^3. If the exponent is negative it is written 3.4E-2 or 3.4e-2. Depending on the device the “E” may be upper or lower case.

Converting units of measurement from one system of measurement to another is a simple process. Usually the only step is correctly multiplying or dividing by a know conversion factor!

First, the units you’re trying to convert between must be of the same type. In most cases length units can only be converted to other length units, area to area, volume to volume, etc. Sometimes units of different types can be related. For example one kilogram was historically equal to one litre of water. Although the relationship between a litre of water and a kilogram is now only an approximation it can be used to estimate a conversion between volume and mass, the catch is the volume must be full of water.

Once you know the desired conversion the first thing you’ll need to find is the conversion factor. In this case we’ll be converting feet to meters, so we could consult the online feet to meters conversion table. After you’ve found the conversion factor apply it by multiplying. We know that we should multiply because in the conversion table the conversion factor is given in terms of 1 (one) foot, the unit we’re trying to convert from. See the steps in action below:

Step 1 – Definition of the Problem:

We want to convert a known length to a length in another system of measurement. In this case we want to convert feet to meters. Both feet and meters are the same type of measurement, they measure the same thing just in different units, we can continue.

Imperial to Metric Conversion: Feet to Meters

Step 2 – Find the Conversion Factor

Using a conversion table find the conversion factor that will convert 1 unit on the left to an unknown number of units on the right side of the expression. In this case the conversion factor for feet to meters is 1 foot = 0.3048 meters.

Imperial to Metric: Conversion Factor

Step 3 – Applying the Conversion Factor

Our conversion factor is written with the known units in mind, 1 ft = . The conversion factor tells us what one foot is equal to so to find out what many feet are equal to we only need to multiple. In the example below 20 feet multiplied by 0.3048 is equal to 6.096 meters. The same conversion factor could be used in the opposite direction but this time we would divide. You can check this by dividing the answer 6.096 (m) by 0.3048 and getting 20 (ft). We know to divide because the conversion formula is not written in terms of one meter. If the conversion formula were written 1 meter = 3.2808399 feet then we could multiple because it is written in terms of one meter. If the conversion formula were written 0.5 feet = 0.1524 meters then we’d have to work on it until one side was equal to one.

Imperial to Metric - Feet to Meters Conversion Factor Applied

To recap, if both units are measuring the same type of thing find the conversion factor written in terms of 1 known unit equals some quantity of another unit and multiply.